# Li-Yau gradient estimates for curvature flows in positively curved manifolds

## TL;DR

This paper establishes new differential Harnack inequalities for convex hypersurface flows in positively curved Einstein manifolds, extending to more general curvature flows and providing monotonicity results in symmetric spaces.

## Contribution

It introduces novel Harnack inequalities for curvature flows with powers of mean curvature in positively curved ambient spaces, under specific curvature bounds.

## Key findings

- Differential Harnack inequalities for convex hypersurface flows in Einstein manifolds.
- New Harnack inequalities for curvature flows in spheres.
- Monotonicity estimate for mean curvature flow in non-negatively curved symmetric spaces.

## Abstract

We prove differential Harnack inequalities for flows of strictly convex hypersurfaces by powers $p$, $0<p<1$, of the mean curvature in Einstein manifolds with a positive lower bound on the sectional curvature. We assume that this lower bound is sufficiently large compared to the derivatives of the curvature tensor of the ambient space and that the mean curvature of the initial hypersurface is sufficiently large compared to the ambient geometry. We also obtain some new Harnack inequalities for more general curvature flows in the sphere, as well as a monotonicity estimate for the mean curvature flow in non-negatively curved, locally symmetric spaces.